301edo

← 300edo

301edo

302edo →

Prime factorization	7 × 43
Step size	3.98671 ¢
Fifth	176\301 (701.661 ¢)
Semitones (A1:m2)	28:23 (111.6 ¢ : 91.69 ¢)
Consistency limit	17
Distinct consistency limit	17

Theory

301edo is a strong 7-limit system, and distinctly consistent through the 17-odd-limit. The equal temperament tempers out 32805/32768 in the 5-limit, 2401/2400 in the 7-limit, 3025/3024, 5632/5625, 8019/8000 in the 11-limit, 729/728, 847/845, 1001/1000, 1716/1715, 2200/2197 in the 13-limit, and 561/560, 833/832, 1089/1088, 1156/1155, 1275/1274 and 1701/1700 in the 17-limit. Since it tempers out both 32805/32768 and 2401/2400, it supports the sesquiquartififths temperament.

Prime harmonics

Approximation of prime harmonics in 301edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-0.29	+0.40	-0.06	-1.15	+0.67	-1.30	+1.49	+1.63	-1.01	-0.85
	Relative (%)	+0.0	-7.4	+10.0	-1.4	-28.9	+16.8	-32.6	+37.4	+40.8	-25.2	-21.3
Steps (reduced)		301 (0)	477 (176)	699 (97)	845 (243)	1041 (138)	1114 (211)	1230 (26)	1279 (75)	1362 (158)	1462 (258)	1491 (287)

Subsets and supersets

Since 301 factors into 7 × 43, 301edo has 7edo and 43edo as its subsets. This is related to the proposal of the deaf French mathematician and acoustician Joseph Sauveur to divide the octave in 43 parts called merides, and those into seven more parts called heptamerides. Back in the days of slide rules and log tables, this made sense since by multiplying the log base ten of the interval in question by 1000, one came close to how many heptamerides it constituted.

301edo also tempers out [168 -43 -43⟩ and 5250987/5242880, so it supports the meridic temperament.

Intervals

Steps	Cents	Approximate ratios	Ups and downs notation
0	0	1/1	D
1	3.99		^D, ^6E♭♭
2	7.97		^^D, ^7E♭♭
3	11.96		^3D, ^8E♭♭
4	15.95		^4D, ^9E♭♭
5	19.93	85/84, 86/85, 87/86, 88/87	^5D, ^10E♭♭
6	23.92	75/74	^6D, ^11E♭♭
7	27.91	63/62, 64/63	^7D, ^12E♭♭
8	31.89	55/54	^8D, ^13E♭♭
9	35.88	48/47, 49/48	^9D, v14E♭
10	39.87	44/43, 87/85	^10D, v13E♭
11	43.85	40/39	^11D, v12E♭
12	47.84	37/36	^12D, v11E♭
13	51.83	34/33	^13D, v10E♭
14	55.81	94/91, 95/92	^14D, v9E♭
15	59.8	88/85	v13D♯, v8E♭
16	63.79		v12D♯, v7E♭
17	67.77	26/25	v11D♯, v6E♭
18	71.76	49/47	v10D♯, v5E♭
19	75.75	47/45	v9D♯, v4E♭
20	79.73		v8D♯, v3E♭
21	83.72	85/81	v7D♯, vvE♭
22	87.71	81/77	v6D♯, vE♭
23	91.69	39/37, 58/55	v5D♯, E♭
24	95.68	37/35, 93/88	v4D♯, ^E♭
25	99.67		v3D♯, ^^E♭
26	103.65	69/65, 86/81	vvD♯, ^3E♭
27	107.64	33/31, 50/47	vD♯, ^4E♭
28	111.63	16/15	D♯, ^5E♭
29	115.61	31/29, 77/72	^D♯, ^6E♭
30	119.6	15/14	^^D♯, ^7E♭
31	123.59	29/27	^3D♯, ^8E♭
32	127.57		^4D♯, ^9E♭
33	131.56	41/38, 68/63	^5D♯, ^10E♭
34	135.55	40/37, 93/86	^6D♯, ^11E♭
35	139.53		^7D♯, ^12E♭
36	143.52	63/58, 88/81	^8D♯, ^13E♭
37	147.51	49/45	^9D♯, v14E
38	151.5		^10D♯, v13E
39	155.48	35/32, 93/85	^11D♯, v12E
40	159.47	34/31, 57/52	^12D♯, v11E
41	163.46		^13D♯, v10E
42	167.44	76/69	^14D♯, v9E
43	171.43	85/77	v13D𝄪, v8E
44	175.42	52/47	v12D𝄪, v7E
45	179.4		v11D𝄪, v6E
46	183.39		v10D𝄪, v5E
47	187.38	39/35	v9D𝄪, v4E
48	191.36	86/77	v8D𝄪, v3E
49	195.35	47/42	v7D𝄪, vvE
50	199.34	46/41, 55/49	v6D𝄪, vE
51	203.32	9/8	E
52	207.31	62/55	^E, ^6F♭
53	211.3	87/77	^^E, ^7F♭
54	215.28	77/68	^3E, ^8F♭
55	219.27	42/37	^4E, ^9F♭
56	223.26	33/29, 58/51, 91/80	^5E, ^10F♭
57	227.24	57/50, 65/57	^6E, ^11F♭
58	231.23	8/7	^7E, ^12F♭
59	235.22	55/48, 63/55	^8E, ^13F♭
60	239.2	31/27	^9E, v14F
61	243.19		^10E, v13F
62	247.18	15/13	^11E, v12F
63	251.16	37/32	^12E, v11F
64	255.15	51/44, 95/82	^13E, v10F
65	259.14	36/31	^14E, v9F
66	263.12		v13E♯, v8F
67	267.11	7/6	v12E♯, v7F
68	271.1	76/65	v11E♯, v6F
69	275.08	34/29, 75/64	v10E♯, v5F
70	279.07	47/40, 74/63	v9E♯, v4F
71	283.06		v8E♯, v3F
72	287.04	85/72	v7E♯, vvF
73	291.03		v6E♯, vF
74	295.02	51/43	F
75	299	82/69	^F, ^6G♭♭
76	302.99	56/47, 81/68	^^F, ^7G♭♭
77	306.98	37/31, 43/36	^3F, ^8G♭♭
78	310.96		^4F, ^9G♭♭
79	314.95		^5F, ^10G♭♭
80	318.94		^6F, ^11G♭♭
81	322.92	47/39	^7F, ^12G♭♭
82	326.91	93/77	^8F, ^13G♭♭
83	330.9	23/19	^9F, v14G♭
84	334.88	91/75	^10F, v13G♭
85	338.87	45/37	^11F, v12G♭
86	342.86	39/32	^12F, v11G♭
87	346.84	11/9	^13F, v10G♭
88	350.83	49/40, 60/49	^14F, v9G♭
89	354.82	27/22	v13F♯, v8G♭
90	358.8	16/13	v12F♯, v7G♭
91	362.79	37/30	v11F♯, v6G♭
92	366.78	68/55	v10F♯, v5G♭
93	370.76	57/46	v9F♯, v4G♭
94	374.75	36/29, 77/62	v8F♯, v3G♭
95	378.74	56/45	v7F♯, vvG♭
96	382.72		v6F♯, vG♭
97	386.71	5/4	v5F♯, G♭
98	390.7	94/75	v4F♯, ^G♭
99	394.68	49/39, 54/43	v3F♯, ^^G♭
100	398.67	34/27	vvF♯, ^3G♭
101	402.66	82/65	vF♯, ^4G♭
102	406.64	43/34	F♯, ^5G♭
103	410.63		^F♯, ^6G♭
104	414.62	47/37	^^F♯, ^7G♭
105	418.6		^3F♯, ^8G♭
106	422.59	60/47	^4F♯, ^9G♭
107	426.58	55/43, 87/68	^5F♯, ^10G♭
108	430.56	50/39	^6F♯, ^11G♭
109	434.55	9/7	^7F♯, ^12G♭
110	438.54	85/66	^8F♯, ^13G♭
111	442.52	31/24	^9F♯, v14G
112	446.51	22/17	^10F♯, v13G
113	450.5	48/37	^11F♯, v12G
114	454.49	13/10	^12F♯, v11G
115	458.47	43/33	^13F♯, v10G
116	462.46	64/49, 81/62	^14F♯, v9G
117	466.45	55/42, 72/55	v13F𝄪, v8G
118	470.43	21/16	v12F𝄪, v7G
119	474.42		v11F𝄪, v6G
120	478.41	29/22	v10F𝄪, v5G
121	482.39	37/28	v9F𝄪, v4G
122	486.38	49/37	v8F𝄪, v3G
123	490.37	69/52, 77/58	v7F𝄪, vvG
124	494.35		v6F𝄪, vG
125	498.34	4/3	G
126	502.33		^G, ^6A♭♭
127	506.31	75/56	^^G, ^7A♭♭
128	510.3	47/35	^3G, ^8A♭♭
129	514.29	35/26, 74/55	^4G, ^9A♭♭
130	518.27	58/43, 85/63	^5G, ^10A♭♭
131	522.26		^6G, ^11A♭♭
132	526.25	42/31	^7G, ^12A♭♭
133	530.23		^8G, ^13A♭♭
134	534.22	49/36, 64/47	^9G, v14A♭
135	538.21	86/63	^10G, v13A♭
136	542.19	93/68	^11G, v12A♭
137	546.18	48/35, 85/62	^12G, v11A♭
138	550.17		^13G, v10A♭
139	554.15	62/45, 95/69	^14G, v9A♭
140	558.14	29/21, 69/50	v13G♯, v8A♭
141	562.13		v12G♯, v7A♭
142	566.11	43/31	v11G♯, v6A♭
143	570.1	57/41	v10G♯, v5A♭
144	574.09	39/28	v9G♯, v4A♭
145	578.07	81/58, 88/63	v8G♯, v3A♭
146	582.06	7/5	v7G♯, vvA♭
147	586.05	87/62	v6G♯, vA♭
148	590.03	45/32	v5G♯, A♭
149	594.02	31/22	v4G♯, ^A♭
150	598.01	65/46	v3G♯, ^^A♭
151	601.99	92/65	vvG♯, ^3A♭
152	605.98	44/31	vG♯, ^4A♭
153	609.97	64/45, 91/64	G♯, ^5A♭
154	613.95	77/54	^G♯, ^6A♭
155	617.94	10/7	^^G♯, ^7A♭
156	621.93	63/44	^3G♯, ^8A♭
157	625.91	56/39	^4G♯, ^9A♭
158	629.9	82/57	^5G♯, ^10A♭
159	633.89	62/43, 75/52	^6G♯, ^11A♭
160	637.87		^7G♯, ^12A♭
161	641.86	42/29	^8G♯, ^13A♭
162	645.85	45/31	^9G♯, v14A
163	649.83		^10G♯, v13A
164	653.82	35/24	^11G♯, v12A
165	657.81		^12G♯, v11A
166	661.79	63/43, 85/58	^13G♯, v10A
167	665.78	47/32, 72/49	^14G♯, v9A
168	669.77	81/55	v13G𝄪, v8A
169	673.75	31/21	v12G𝄪, v7A
170	677.74		v11G𝄪, v6A
171	681.73	43/29	v10G𝄪, v5A
172	685.71	52/35, 55/37	v9G𝄪, v4A
173	689.7	70/47	v8G𝄪, v3A
174	693.69		v7G𝄪, vvA
175	697.67		v6G𝄪, vA
176	701.66	3/2	A
177	705.65		^A, ^6B♭♭
178	709.63		^^A, ^7B♭♭
179	713.62	74/49, 77/51	^3A, ^8B♭♭
180	717.61	56/37	^4A, ^9B♭♭
181	721.59	44/29, 91/60	^5A, ^10B♭♭
182	725.58		^6A, ^11B♭♭
183	729.57	32/21	^7A, ^12B♭♭
184	733.55	55/36, 84/55	^8A, ^13B♭♭
185	737.54	49/32, 75/49	^9A, v14B♭
186	741.53	66/43	^10A, v13B♭
187	745.51	20/13	^11A, v12B♭
188	749.5	37/24	^12A, v11B♭
189	753.49	17/11	^13A, v10B♭
190	757.48	48/31	^14A, v9B♭
191	761.46		v13A♯, v8B♭
192	765.45	14/9	v12A♯, v7B♭
193	769.44	39/25	v11A♯, v6B♭
194	773.42	86/55	v10A♯, v5B♭
195	777.41	47/30	v9A♯, v4B♭
196	781.4		v8A♯, v3B♭
197	785.38	74/47, 85/54	v7A♯, vvB♭
198	789.37		v6A♯, vB♭
199	793.36	68/43, 87/55	v5A♯, B♭
200	797.34	65/41	v4A♯, ^B♭
201	801.33	27/17	v3A♯, ^^B♭
202	805.32	43/27, 78/49	vvA♯, ^3B♭
203	809.3	75/47, 91/57	vA♯, ^4B♭
204	813.29	8/5	A♯, ^5B♭
205	817.28	93/58	^A♯, ^6B♭
206	821.26	45/28	^^A♯, ^7B♭
207	825.25	29/18	^3A♯, ^8B♭
208	829.24	92/57	^4A♯, ^9B♭
209	833.22	55/34	^5A♯, ^10B♭
210	837.21	60/37	^6A♯, ^11B♭
211	841.2	13/8	^7A♯, ^12B♭
212	845.18	44/27	^8A♯, ^13B♭
213	849.17	49/30, 80/49	^9A♯, v14B
214	853.16	18/11	^10A♯, v13B
215	857.14	64/39	^11A♯, v12B
216	861.13	74/45	^12A♯, v11B
217	865.12		^13A♯, v10B
218	869.1	38/23	^14A♯, v9B
219	873.09		v13A𝄪, v8B
220	877.08	78/47	v12A𝄪, v7B
221	881.06		v11A𝄪, v6B
222	885.05		v10A𝄪, v5B
223	889.04		v9A𝄪, v4B
224	893.02	62/37, 72/43	v8A𝄪, v3B
225	897.01	47/28	v7A𝄪, vvB
226	901	69/41	v6A𝄪, vB
227	904.98	86/51	B
228	908.97	93/55	^B, ^6C♭
229	912.96		^^B, ^7C♭
230	916.94		^3B, ^8C♭
231	920.93	63/37, 80/47	^4B, ^9C♭
232	924.92	29/17	^5B, ^10C♭
233	928.9	65/38	^6B, ^11C♭
234	932.89	12/7	^7B, ^12C♭
235	936.88		^8B, ^13C♭
236	940.86	31/18	^9B, v14C
237	944.85	88/51	^10B, v13C
238	948.84	64/37	^11B, v12C
239	952.82	26/15	^12B, v11C
240	956.81		^13B, v10C
241	960.8	54/31	^14B, v9C
242	964.78		v13B♯, v8C
243	968.77	7/4	v12B♯, v7C
244	972.76		v11B♯, v6C
245	976.74	51/29, 58/33	v10B♯, v5C
246	980.73	37/21	v9B♯, v4C
247	984.72		v8B♯, v3C
248	988.7	85/48	v7B♯, vvC
249	992.69	55/31	v6B♯, vC
250	996.68	16/9	C
251	1000.66	41/23	^C, ^6D♭♭
252	1004.65	84/47	^^C, ^7D♭♭
253	1008.64	77/43	^3C, ^8D♭♭
254	1012.62	70/39	^4C, ^9D♭♭
255	1016.61		^5C, ^10D♭♭
256	1020.6		^6C, ^11D♭♭
257	1024.58	47/26	^7C, ^12D♭♭
258	1028.57		^8C, ^13D♭♭
259	1032.56	69/38	^9C, v14D♭
260	1036.54	91/50	^10C, v13D♭
261	1040.53	31/17	^11C, v12D♭
262	1044.52	64/35	^12C, v11D♭
263	1048.5		^13C, v10D♭
264	1052.49	90/49	^14C, v9D♭
265	1056.48	81/44	v13C♯, v8D♭
266	1060.47		v12C♯, v7D♭
267	1064.45	37/20	v11C♯, v6D♭
268	1068.44	63/34, 76/41	v10C♯, v5D♭
269	1072.43		v9C♯, v4D♭
270	1076.41	54/29	v8C♯, v3D♭
271	1080.4	28/15	v7C♯, vvD♭
272	1084.39	58/31	v6C♯, vD♭
273	1088.37	15/8	v5C♯, D♭
274	1092.36	47/25, 62/33	v4C♯, ^D♭
275	1096.35	81/43	v3C♯, ^^D♭
276	1100.33		vvC♯, ^3D♭
277	1104.32	70/37	vC♯, ^4D♭
278	1108.31	55/29, 74/39	C♯, ^5D♭
279	1112.29		^C♯, ^6D♭
280	1116.28		^^C♯, ^7D♭
281	1120.27		^3C♯, ^8D♭
282	1124.25	90/47	^4C♯, ^9D♭
283	1128.24	94/49	^5C♯, ^10D♭
284	1132.23	25/13	^6C♯, ^11D♭
285	1136.21		^7C♯, ^12D♭
286	1140.2	85/44	^8C♯, ^13D♭
287	1144.19	91/47	^9C♯, v14D
288	1148.17	33/17	^10C♯, v13D
289	1152.16	72/37	^11C♯, v12D
290	1156.15	39/20	^12C♯, v11D
291	1160.13	43/22	^13C♯, v10D
292	1164.12	47/24	^14C♯, v9D
293	1168.11		v13C𝄪, v8D
294	1172.09	63/32	v12C𝄪, v7D
295	1176.08		v11C𝄪, v6D
296	1180.07	85/43, 87/44	v10C𝄪, v5D
297	1184.05		v9C𝄪, v4D
298	1188.04		v8C𝄪, v3D
299	1192.03		v7C𝄪, vvD
300	1196.01		v6C𝄪, vD
301	1200	2/1	D

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[-477 301⟩	[⟨301 477]]	+0.0927	0.0927	2.33
2.3.5	32805/32768, [3 45 -32⟩	[⟨301 477 699]]	+0.0048	0.1456	3.65
2.3.5.7	2401/2400, 32805/32768, 1959552/1953125	[⟨301 477 699 845]]	+0.0085	0.1262	3.17
2.3.5.7.11	2401/2400, 3025/3024, 5632/5625, 8019/8000	[⟨301 477 699 845 1041]]	+0.0734	0.1720	4.31
2.3.5.7.11.13	729/728, 847/845, 1001/1000, 1716/1715, 3025/3024	[⟨301 477 699 845 1041 1114]]	+0.0310	0.1834	4.60
2.3.5.7.11.13.17	561/560, 729/728, 833/832, 847/845, 1001/1000, 1089/1088	[⟨301 477 699 845 1041 1114 1230]]	+0.0721	0.1973	4.95

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	25\301	99.67	18/17	Quintaschis
1	44\301	175.42	448/405	Sesquiquartififths / sesquart (301e)
1	68\301	271.10	90/77	Quasiorwell
1	76\301	302.99	25/21	Quinmite
1	125\301	498.34	4/3	Helmholtz
7	125\301 (4\301)	498.34 (15.95)	4/3 (245/243)	Septant
43	125\301 (1\301)	498.34 (3.99)	4/3 (540/539)	Meridic

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct